Exponential Functions & Growth

Linear growth adds the same amount each step. Exponential growth multiplies by the same amount — and always wins in the end.

A king offers a reward: one grain of rice on the first square of a chessboard, doubling on each square after.

It sounds modest. One, two, four, eight — you could carry the first dozen squares in a cup.

one grain on the first square, then double every time 1248163264128 square 8 already needs 128 grains, and the board has 64 squares

By square 8 you are at 128 grains. Adding one grain each time would have reached 8.

By square 20 it is half a million. By square 64 the pile is larger than a mountain — more rice than has ever been grown.

And nothing dramatic happened at any single step. Every step was just "times two."

That is exponential growth: it looks slow at the start and then suddenly does not. Linear growth adds a constant; exponential growth multiplies by one — and multiplying always wins in the end.

Linear growth against exponential growth adding the same amount, against multiplying by the same amount + 6 each step x 2 each step time amount linear: y = mx + b exponential: y = a(b)^x the exponential starts behind and still overtakes, every single time

Spotting exponential behaviour

In a table with evenly spaced x values: if consecutive y values have a constant difference, it is linear. If they have a constant ratio, it is exponential.

3, 6, 12, 24 has ratio 2 every time, so it is exponential with b = 2.

3, 6, 9, 12 has difference 3 every time, so it is linear.

Differences or ratios

Subtract to test for linear. Divide to test for exponential. Do both on any unfamiliar table and one of them will come out constant.

Percentages and the growth factor

Growing by 8% means multiplying by 1.08, because you keep the original 100% and add 8% more.

Shrinking by 8% means multiplying by 0.92.

So b = 1 + r for growth and b = 1 - r for decay, with r written as a decimal.

Do not use 0.08 as the base

Multiplying by 0.08 destroys 92% of the amount each step. Growth of 8% needs the factor 1.08.

The graph

Growth curves rise slowly at first and then steeply — always increasing, always concave up.

Decay curves fall fast and then level off, approaching the x-axis without ever touching it.

The y-intercept is always a, since b0 = 1 whatever b is.

Check x = 0 first

Whatever the base, b0 = 1, so f(0) = a. That gives you one point of the graph with no work at all.

Linear against exponential

Exponential growth eventually overtakes any linear growth, however gentle the base or steep the line.

Over small ranges a linear model may fit exponential data acceptably, which is why short-term data can mislead.

Doubling every step from 1 reaches a million in twenty steps. Adding one reaches twenty.

Geometric sequences

A geometric sequence has a constant ratio: an = a1 · rn-1.

It is an exponential function restricted to the whole numbers.

Recursively: an = r · an-1, starting from a1.

Worked examples

Table or not

Is 5, 15, 45, 135 linear or exponential? Write the rule.

  1. Differences: 10, 30, 90 — not constant. So not linear.
  2. Ratios: 155 = 3, 4515 = 3, 13545 = 3 Constant ratio, so exponential.
  3. At x = 0 the value would be 5, so a = 5 and b = 3. Treating 5 as the initial term.
  4. f(x) = 5 · 3x Check f(2) = 5(9) = 45 ✓

f(x) = 5 · 3x

Percentage growth

A town of 12,000 grows 3% a year. Write a model and find the population after 10 years.

  1. a = 12000 and b = 1 + 0.03 = 1.03 Keep the original, add 3%.
  2. P(t) = 12000(1.03)t With t in years.
  3. P(10) = 12000(1.03)10 ≈ 12000(1.3439) Raise to the tenth power.
  4. ≈ 16,127 people Sensible: about a third more after a decade.

P(t) = 12000(1.03)t; about 16,127

Decay

A car worth $24,000 loses 15% of its value each year. What is it worth after 4 years?

  1. Losing 15% leaves 85%, so b = 0.85. 1 - 0.15 = 0.85.
  2. V(t) = 24000(0.85)t Exponential decay.
  3. V(4) = 24000(0.85)4 = 24000(0.52200625) Four years of shrinking.
  4. ≈ $12,528 A little over half, which matches losing 15% four times.

About $12,528

Words to know

Exponential function
f(x) = a · bx with a ≠ 0, b > 0, b ≠ 1.
Growth factor
b itself. b > 1 means growth, 0 < b < 1 means decay.
Initial value
a, the output when x = 0.
Growth rate
The percentage change. b = 1 + r for growth, b = 1 - r for decay.
Asymptote
A line the graph approaches but never reaches — here, the x-axis.

Mistakes to watch for

Using the percentage as the base. Growth of 8% means multiplying by 1.08, because you keep the original 100% and add 8 more. Multiplying by 0.08 destroys 92% of the amount every step.

Testing only for a constant difference. If the differences are not constant, try the ratios — a constant ratio means exponential. Checking only one of the two makes exponential data look like an error.

Multiplying the base by the exponent. 3 · 24 is 3 × 16 = 48, not 64. The exponent attaches to the 2 alone, and the 3 is a separate multiplier waiting its turn.

Believing a decay curve reaches zero. Halving again and again gets arbitrarily small but never arrives. That is what an asymptote is, and it is why a drug never completely leaves the body.

The rice-and-chessboard story appears in Indian and Persian sources from at least the 11th century, usually ending with the king realising he cannot pay.

The total is 264 - 1 grains: about 18 quintillion, roughly a thousand years of world rice production.

The same arithmetic explains compound interest, viral spread and the growth of computing power — one idea, wearing very different clothes.

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